A Simple Derivation of Newton-Cotes Formulas with Realistic Errors

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1 Journal of Mathematics Research; Vol. 4, No. 5; 01 ISSN E-ISSN Published by Canadian Center of Science and Education A Simple Derivation of Neton-Cotes Formulas ith Realistic Errors Mário M. Graça 1 1 Department of Mathematics, Instituto Superior Técnico, Technical University of Lisbon, Lisboa, Portugal Correspondence: Mário M. Graça, Department of Mathematics, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, Lisboa , Portugal Received: June 8, 01 Accepted: July, 01 Online Published: September 11, 01 doi:10.559/jmr.v4n5p4 URL: Abstract In order to approximate the integral I( f ) = b f (x)dx, here f is a sufficiently smooth function, models for a quadrature rules are developed using a given panel of n (n ) equally spaced points. These models arise from the undetermined coefficients method, using a Neton s basis for polynomials. Although part of the final product is algebraically equivalent to the ell knon closed Neton-Cotes rules, the algorithms obtained are not the classical ones. In the basic model the most simple quadrature rule Q n is adopted (the so-called left rectangle rule) and a correction Ẽ n is constructed, so that the final rule S n = Q n + Ẽ n is interpolatory. The correction Ẽ n, depending on the divided differences of the data, might be considered a realistic correction for Q n, in the sense that Ẽ n should be close to the magnitude of the true error of Q n, having also the correct sign. The analysis of the theoretical error of the rule S n as ell as some classical properties for divided differences suggest the inclusion of one or to ne points in the given panel. When n is even it is included one point and to points otherise. In both cases this approach enables the computation of a realistic error Ē S n for the extended or corrected rule S n. The respective output (Q n, Ẽ n, S n, Ē S n ) contains reliable information on the quality of the approximations Q n and S n, provided certain conditions involving ratios for the derivatives of the function f are fulfilled. These simple rules are easily converted into composite ones. Numerical examples are presented shoing that these quadrature rules are useful as a computational alternative to the classical Neton-Cotes formulas. Keyords: divided differences, undetermined coefficients method, realistic error, Neton-Cotes rules 1. Introduction The first to quadrature rules taught in any numerical analysis course belong to a group knon as closed Neton- Cotes rules. They are used to approximate the integral I( f ) = b f (x)dx ofasufficiently smooth function f in the a finite interval [a, b]. The basic rules are knon as trapezoidal rule and the Simpson s rule. The trapezoidal rule is Q( f ) = h/(f (a) + f (b)), for hich h = b a, and has the theoretical error I( f ) Q( f ) = h 1 f () (ξ), (1) hile the Simpson s rule is Q( f ) = h/(f (a) + 4 f ((a + b)/) + f (b)), ith h = (b a)/, and its error is I( f ) Q( f ) = h5 90 f (4) (ξ). () The error formulas (1) and () are of existential type. In fact, assuming that f () and f (4) are (respectively) continuous, the expressions (1) and () say that there exist a point ξ, somehere in the interval (a, b), for hich the respective error has the displayed form. From a computational point of vie the utility of these error expressions is rather limited since in general is quite difficult or even impossible to obtain expressions for the derivatives f () or f (4), and consequently bounds for I( f ) Q( f ). Even in the case one obtains such bounds they generally overestimate the true error of Q( f ). Under mild assumption on the smoothness of the integrand function f, our aim is to determine certain quadrature rules, say R( f ), as ell as approximations for its error Ẽ( f ), using only the information contained in the table of 4

2 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 values or panel arising from the discretization of the problem. The algorithm to be constructed ill produce the numerical value R( f ), the correction or estimated error Ẽ( f ) as ell as the value of the interpolatory rule S ( f ) = R( f ) + Ẽ( f ). () The true error of S ( f ) should be much less than the estimated error of R( f ), that is, I( f ) S ( f ) << Ẽ( f ), (4) for a sufficiently small step h. In such case e say that Ẽ( f )isarealistic correction for R( f ). Unlike the usual approach here one builds a quadrature formula Q( f ) (like the trapezoidal or Simpson s rule) hich is supposed to be a reasonable approximation to the exact value of the integral, here e do not care ether the approximation R( f ) is eventually bad, provided that the correction Ẽ( f ) has been ell modeled. In this case the value S ( f ) ill be a good approximation to the exact value of the integral I( f ). Besides the values R( f ), Ẽ( f ) and S ( f ) e are also interested in computing a good estimation Ē S ( f ) for the true error of S ( f ), in the folloing sense. If the true error E( f ) = I( f ) S ( f ) is expressed in the standard decimal form as E( f ) = ±0.d 1 d d m 10 k, k 0, the approximation Ē( f ) is said to be realistic if its decimal form has the same sign as E( f ) and its first digit in the mantissa differs at most one unit, that is, Ē( f ) = ±0.(d 1 ± 1) 10 k (the dots represent any decimal digit). Finally, the algorithm to be used ill produce the values (R( f ), Ẽ( f ), S ( f ), Ē( f )). In section 1.1 e present to models for building simple quadrature rules named model A and model B. Although both models are derived from the same method, in this ork e focus our attention mainly on the model A. Definitions, notations and background material are presented in section. In Proposition e obtain the eights for the quadrature rule in model A by the undetermined coefficients method as ell as the theoretical error expressions for the rules are deduced (see Proposition 4). The main results are discussed in Section, namely in Proposition 11 e sho that a reliable computation of realistic errors depends on the behavior of a certain function involving ratios beteen high order derivatives of the integrand function f and its first derivative. Composite rules for model A are presented in Section 4 here some numerical examples illustrate ho our approach allos to obtain realistic error s estimates for these rules. 1.1 To Models In this ork e consider to be given a panel of n (n ) points {(x 1, f 1 ), (x, f ),...,(x n, f n )}, in the interval [a, b], having the nodes x i equally spaced ith step h > 0, f i = f (x i ), here f asufficiently smooth function in the interval. We consider the folloing to models: Model A Using only the first node of the panel e construct a quadrature rule Q n ( f ) adding a correction Ẽ n ( f ), so that the corrected or extended rule S n ( f ) = Q n ( f ) + Ẽ n ( f )isinterpolatory for the hole panel, S n ( f ) = Q n ( f ) + Ẽ n ( f ) = a 1 f (x 1 ) + {a f [x 1, x ] + + a n f [x 1, x,...,x n ]}, (5) here f [x 1, x,...,x n ] denotes the (n 1)-th divided difference and a 1, a,..., a n are eights to be determined. Note that Q n ( f ) is simply the so-called left rectangle rule, thus Ẽ n ( f ) = n j= a j f [x 1,...,x j ] can be seen as a correction to such a rule. Model B The rule Q n ( f ) uses the first n 1 points of the panel (therefore is not interpolatory in the hole panel), and it is added a correction term Ẽ n ( f ), so that the corrected or extended rule S n ( f ) is interpolatory, S n ( f ) = Q n ( f ) + Ẽ n ( f ) = { a 1 f (x 1 ) + a f (x ) + + a n 1 f (x n 1 ) } + a n f [x 1, x,...,x n ]. (6) Since the interpolating polynomial of the panel is unique, the value computed for S n ( f ) using either model is the same and equal to the value one finds if the simple closed Neton-Cotes rule for n equally spaced points has been applied to the data. This means that the extended rules (5) and (6) are both algebraically equivalent to the referred simple Neton-Cotes rules. Hoever, the algorithms associated to each of the models (5) and (6) are not the 5

3 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 classical ones for the referred rules. In particular, e can sho that that for n odd, the rules Q n ( f ) in model B are open Neton-Cotes formulas (Graça, 01). Therefore, the extended rule S n in model B can be seen as a bridge beteen open and closed Neton-Cotes rules. The method of undetermined coefficients applied to a Neton s basis of polynomials is used in order to obtain S n ( f ). The associated system of equations is diagonal, The same method can also be applied to get any hybrid model obtained from the models A and B. For instance, an hybrid extended rule using n = points could be ritten as S ( f ) = {a 1 f (x 1 ) + a f (x ) + a f (x )} + a 4 f [x 1, x ] + a 5 f [x 1, x, x ]. In this ork our study is mainly focused in model A.. Notation and Background Definition 1 (Canonical and Neton s basis) Let P k be the vector space of real polynomials of degree less or equal to k (k a nonnegative integer). The set <φ 0 (x),φ 1 (x),...,φ n 1 (x) >, here φ i (x) = x i, i = 0, 1,...,n 1, (7) is the canonical basis for P n 1. Given n 1 distinct points x 1, x,...x n, the set < 0 (x), 1 (x),..., n 1 (x) >, ith 0 (x) = 1 i (x) = i 1 (x) (x x i ), i = 1,...,(n 1) (8) is knon as the Neton s basis for P n 1. A polynomial interpolatory quadrature rule R n ( f ) obtained from a given panel {(x 1, f 1 ), (x, f ),...,(x n, f n )}, here x i x j for i j, has the form R n ( f ) = c 1 f (x 1 ) + c f (x ) + + c n f (x n ), (9) here the coefficients (or eights) c j (1 j n) can be computed assuming the quadrature rule is exact for any polynomial q of degree less or equal to n 1, that is, deg(r n ( f )) = n 1, according to the folloing definition. Definition (Degree of exactness) (Gautschi, 1997, p. 157) A quadrature rule R n ( f ) = n i=1 c i f (x i ) has (polynomial) degree of exactness d if the rule is exact henever f is a polynomial of degree d, that is E n ( f ) = I( f ) R n ( f ) = 0, for all f P d. The degree of the quadrature rule is denoted by deg(r). When deg(r) = n 1 the rule is called interpolatory. In particular for a n-point panel the interpolating polynomial p satisfies f (x) = p(x) + r(x), x [a, b], (10) here p can be ritten in Neton s form (see for instance Steffensen, 006, p., or any standard text in numerical analysis) p(x) = f 1 + f [x 1, x ](x x 1 ) + + f [x 1, x,...,x n ](x x 1 )(x x ) (x x n 1 ) (11) and the remainder-term is r(x) = f [x 1, x,...,x n, x](x x 1 )(x x ) (x x n ), (1) here f i = f (x i ) and f [x 1, x,...,x 1+ j ], for j 0, denotes the j-th divided difference of the data (x i, f i ), ith i = 1,...,n. Therefore from (10) e obtain I( f ) = b a p(x)dx + b here E Rn ( f ) denotes the true error of the rule R n ( f ). Thus, R n ( f ) = xn x 1 0 (x)dx f 1 + xn a x 1 1 (x)dx f [x 1, x ] + + r(x)dx = R n ( f ) + E Rn ( f ), (1) 6 xn x 1 n 1 (x)dx f [x 1, x,...,x n ] (14)

4 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 The expression (14) suggests that the application of the undetermined coefficient method using the Neton s basis for polynomials should be rearding since the successive divided differences are trivial for such a basis. In particular, the eights for the extended rule in model A are trivially computed. Proposition The eights for the rule S n ( f ) in model A are a i = I( i 1 ) = (n 1) h 0 i 1 (t)dt i = 1,,...,n (15) Proof. The divided differences do not depend on a particular node but on the distance beteen nodes. Thus for any given n-point panel of constant step h, e can assume ithout loss of generality that x 1 = 0, x = h,..., x n = (n 1)h. Considering the Neton s basis for polynomials 0 (t) = 1, 1 (t) = t,..., n 1 (t) = t(t h) (t (n )h), here 0 t (n 1) h, from (5) e have S n ( 0 ) = a 1, S n ( 1 ) = a,...,s n ( i ) = a i+1 for i = 0,...,(n 1). The undetermined coefficients method applied to the Neton s basis < 0 (t), 1 (t),..., n 1 (t) > leads to the n conditions a i = I( i 1 ) or, equivalently, to a diagonal system of linear equations hose matrix is the identity. The equalities in (15) can also be obtained directly from (14). Theoretical expressions for the error E Rn ( f ) in (1) can be obtained either via the mean value theorem for integrals or by considering the so-called Peano kernel (Davis & Rabinoitz, 1984, p. 85; Gautschi, 1997, p. 176). Hoever, e ill use the method of undetermined coefficients henever theoretical expressions for the errors E Qn and E S n are needed. For sufficiently smooth functions f, the fundamental relationship beteen divided differences on a given panel and the derivatives of f is given by the folloing ell knon result, Proposition 4 (Steffensen, 006, p. 4; Gautschi, 1997, p. 101; Krylov, 005, p. 41) Given n (n ) distinct nodes {x 1,...,x n } in J = [a, b], and f C n 1 (J), there exists ξ (x 1, x n ) such that f [x 1, x,...,x n ] = f (n 1) (ξ) (n 1)!. (16) Applying (16) to the canonical or Neton s basis, e get φ j [x 1, x,...,x n ] = j [x 1, x,...,x n ] = 0, for j = 0, 1,...,(n ) φ n 1 [x 1, x,...,x n ] = n 1 [x 1, x,...,x n ] = 1. (17) By construction, the rules S n in models A and B are at least of degree n 1 of precision according to Definition. In this ork the undetermined coefficients method enables us to obtain both the eights and theoretical error formulas. This apparently contradicts the folloing assertion due to Walter Gautschi (p. 176): The method of undetermined coefficients, in contrast, generates only the coefficients in the approximation and gives no clue as to the approximation error. Note that by Definition the theoretical error (1) says that deg(q) = 1 for the trapezoidal rule, and from () one concludes that deg(q) = for the Simpson s rule. This suggests the folloing assumption. Assumption 5 Let be given a n-point (n 1) panel ith constant step h > 0, asufficiently smooth function f defined on the interval [a, b], and a quadrature rule R( f ) (interpolatory or not) of degree m, there exists a constant K h 0 (depending on a certain poer of h) and a point ξ, such that E( f ) = I( f ) R( f ) = K h f (m+1) (ξ), ξ (a, b) (18) here de derivative f (m+1) is not identically null in [a, b], and m is the least integer for hich (18) holds. The expression (18) is crucial in order to deduce formulas for the theoretical error of the rules in model A or B. Proposition 6 Under Assumption 5, the constant K h in (18) is K h = I(ω m+1) R(ω m+1 ), (19) (m + 1)! 7

5 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 here ω m+1 (x) is either the element φ m+1 (x) of the canonical basis, or the element m+1 (x) of the Neton s basis, or any polynomial of degree m+1 taken from any basis for polynomials used to apply the undetermined coefficients method. Proof. For any nonnegative integer i m, from (18) e get E(ω i ) = I(ω i ) R(ω i ) = K h 0. As m is the least integer for hich the righthand side of (18) is non zero, and deg(r) = m, one has I(ω m+1 ) R(ω m+1 ) = K h ω (m+1) m+1 (x) = K h (m + 1)!, from hich it follos (19). As for a fixed basis the interpolating polynomial is unique, it follos that the error for the corresponding interpolatory rule is unique as ell. In Proposition 8 e sho that deg(s n ( f )) depends on the parity of n. Thus, e recover a ell knon result about the precision of the Neton-Cotes rules, since S n ( f ) is algebraically equivalent to a closed Neton Cotes formula ith n nodes. Let us first prove the folloing lemma. Lemma 7 Consider the Neton s polynomials 0 (t) = 1 1 (t) = t j (t) = j 1 i=1 t (t ih), j. Let n 1 be an integer and I [ 1], I [0] and I [ ] the folloing integrals: I [ 1] ( n ) = (n 1) h 0 n (t)dt I [0] ( n ) = nh 0 n (t)dt I [ ] ( n ) = (n ) h 0 n (t)dt. Then, (a) I [ 1] ( n ) 0 for n even and I [ 1] ( n ) = 0 for n odd; (b) I [0] ( n ) 0 for n and I [ ] ( n ) 0 for n. Proof. (a) For n = 1 it is obvious that I [ 1] (1) = 0. For any integer n, let us change the integration interval [0, (n 1) h] into the interval I = [ (n 1)/, (n 1)/] and consider the bijection x = ω(t) = h (x + (n 1)/). For n odd, e obtain (n 1) h I [ 1] ( n ) = n (t)dt 0 (n 1)/ ( ) ( n 1 n 1 = h n+1 t t 1) (n 1)/ ( n 1 t ) (t 1 ) t dt. As the integrand is an odd function in I,ehaveI [ 1] ( n ) = 0. For n even, e get (n 1) h I [ 1] ( n ) = n (t)dt 0 (n 1)/ = h [(t n+1 (n 1)/ (n 1) )(t 4 ) (n ) 4 (t ) ] (n 5) (t 1/4) dt, 4 here the integrand is an even function, thus I [ 1] ( n ) 0. (b) The proof is analogous so it is ommited. The degree of precision for the rules in models A and B, and the respective true errors can be easily obtained using the undetermined coefficients method, the Lemma 7 and Proposition 6. The next propositions (8, 9 and 10) 8

6 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 establish the theoretical errors and degree of precision of these rules. In particular, in Proposition 8 e recover a classical result on the theoretical error for the rule S n ( f ) - see for instance Isaacson and Keller (1966, p. 1). Proposition 8 Consider the rule S n ( f ) for the models A or B defined in the panel {(x 1, f 1 ),...,(x n, f n )}, and assume that J = [a, b] is a finite interval containing the nodes x 1,...,x n, for n. Let n (x) denote the Neton s polynomial of degree n. The respective degree of precision and true error E S n = I( f ) S n ( f ) are the folloing: (i) If n is odd and f C n+1 (J), then (ii) If n is even and f C n (J), then deg(s n ) = n E S n = I( n+1) (n + 1)! f (n+1) (ξ), ξ J. (0) deg(s n ) = n 1 E S n = I( n) f (n) (ξ), ξ J. n! Proof. We can assume ithout loss of generality that the panel is {(0, f 1 ), (h, f ),...,((n 1)h, f n )} (just translate the point x 1 ).Forn even or odd, by construction of the interpolatory rule S n ( f ), e have deg(s n ) n 1 in model A or B. Taking the Neton s polynomial n (t) = t(t h) (t (n 1)h)), hose zeros are 0, h,...,(n 1) h, e get for the divided differences in (5), n (0) = n [0, h] =...= n [0, h,...,(n 1)h] = 0, here f has been substituted by n in (5). Thus, S n ( n ) = 0 and S j ( n ) = 0 for any j n + 1. (i) For n odd, by Lemma 7 (a) e get I( n ) = 0. Thus S n ( n ) = I( n ), and so deg(s n ) n + 1. Hoever, by Lemma 7 (b), e have I( n+1 ) = (n 1) h 0 n+1 (t)dt 0. Therefore, deg(s n ) = n and so (0) follos by Proposition 6. Proposition 9 Consider the rule Q n ( f ) = I( 0 ) f (x 1 ) given by model A, and assume that f C(J), here J = [a, b] is a finite interval containing the nodes x 1, x,...,x n, for n. Then, there exists a point ξ J such that E Qn ( f ) = I( f ) Q n ( f ) = I( 1 ) f (ξ) = (n 1) h f (ξ). () Proof. Taking x 1 = 0 and 1 (t) = t, ehaveq n ( 1 ) = 0 and I( 1 ) 0. Thus, by Proposition 6, e obtain the equalities in (). Proposition 10 Consider the rule Q n ( f ) given in model B and assume that f C n (J), here J = [a, b] is a finite interval containing the nodes x 1,...,x n, for n. Let n 1 (x) be the Neton s polynomial of degree n 1. The degree of precision for Q n is n and there exists a point ξ J such that E Qn ( f ) = I( n 1) (n 1)! f (n 1) (ξ). () Proof. Without loss of generality consider the panel {(0, f 1 ), (h, f ),...,((n 1)h, f n )}. By construction, via the undetermined coefficients, e have deg(q n ) (n ). Taking n 1 (t) = t (t h)...(t (n )h), e have n 1 (t i ) = 0, for i = 0,...,(n ), so Q( (n 1) ) = 0. By Lemma 7 (b), I( n 1 ) = (n 1)h 0 n 1 (t)dt 0, and therefore m = deg(q ) = n. Thus, by Proposition 4, there exist θ (0, (n 1)h) such that, E Qn ( f ) = I( f ) Q n ( f ) = I( n 1) Q n 1 (m + 1)! f (m+1) (θ) = I( n 1) (n 1)! f (n 1) (θ). To the point θ it corresponds a point ξ in the interval J, and so () holds.. Realistic Errors for Model A The properties discussed in the previous Section are valid for both models A and B. Hoever here e ill only present some numerical examples for the rules defined by model A. A detailed discussion and examples for model B ill be presented elsehere. (1) 9

7 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 From (15) e obtain immediately the eights for any rule of n points defined by model A. Such eights are displayed in Table 1, for n 9. The values displayed should be multiplied by an appropriate poer of h as indicated in the table s label. According to Proposition 8, the last column in this table contains the value d = deg(s n ) for the degree of precision of the rule S n ( f ). Table 1. Weights for model A, a j = (n 1)h 0 j 1 (t)dt, j = 1,,...,n. The entries in column (heading a 1 ) should be multiplied by h; the entries in column (heading a ) multiplied by h, and so on n a 1 a a a 4 a 5 a 6 a 7 a 8 a 9 d Note that, by construction, the eights in model A are positive for any n. Therefore, the respective extended rule S n ( f ) does not suffers from the inconvenient observed in the traditional form for Neton-Cotes rules here, for n large (n 9) the eights are of mixed sign leading eventually to losses of significance by cancellation. The next Proposition 11 shos that a reliable computation of realistic errors for the rule S n ( f ), for n, depends on the behavior of a certain function g(x, h) involving certain quotients beteen derivatives of higher order of f and its first derivative. Fortunately, in the applications, only a crude information on the function g(x, h) is needed, and in practice it ill be sufficient to plot g(x, h) for some different values of the step h, as it is illustrated in the numerical examples given in this Section (for some simple rules) and in Section 4 (for some composite rules). Proposition 11 Consider a panel of n points and the model A for approximating I( f ) = b f (x)dx, here f is a asufficiently smooth function defined in the interval J = [a, b] containing the panel nodes. Let S n ( f ) = Q n ( f ) + Ẽ n ( f ) = a 1 f (x 1 ) + { n k= a k f [x 1, x,...x k ] }, here a i = I( i 1 ), i = 1,,...,n. Denote by g(x, h) (or g(x) hen h is fixed) the function n 1 g(x, h) = 1 + a j+1 f ( j) (x) a j! f (x). (4) Assuming that and j= f (x) 0 x (x 1, x n ) (5) g(x, h) h x (x 1, x n ) (6) then, for a sufficiently small step h > 0, the correction Ẽ n ( f ) is realistic for Q n ( f ). Furthermore, the true error of S n ( f ) can be estimated by the folloing realistic errors: (a) For n odd: Ē S n = I( n+1) I( 1 ) here x 1 = (x 1 + x )/ and x = (x n 1 + x n )/. f [x 1, x,...,x n, x 1, x ] f [x 1, x ] Ẽ n ( f ), (7) 40

8 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 (b) For n even: Ē S n = I( n) I( 1 ) here x = (x 1 + x )/. Proof. (a) By Proposition 8 (i), e have f [x 1, x,...,x n, x] f [x 1, x ] I( f ) S n ( f ) = I( n+1) (n + 1)! f (n+1) (ξ), ξ (x 1, x n ), and from Proposition 4 the correction Ẽ n ( f ) can be ritten as Ẽ n ( f ) = a f (θ) + a f () (θ 1 )! Ẽ n ( f ), (8) f (n 1) (θ n ) a n, (n 1)! here θ (x 1, x ), θ 1 (x 1, x ),..., θ n (x 1, x,...,x n ). Therefore, ( Ẽ n ( f ) = a f (θ) 1 + n 1 a j+1 f ( j) ) (θ j 1 ) j= ( a j! f (θ) = I( 1 ) f (θ) 1 + n 1 j= Thus, using the hypothesis in (5) e obtain that is, I( f ) S n ( f ) I( n+1 ) = Ẽ n ( f ) (n + 1)! I( 1 ) I( f ) S n ( f ) Ẽ n ( f ) = chn (n + 1)! f (n+1) (ξ) f (θ) f (n+1) (ξ) f (θ) a j+1 f ( j) (θ j 1 ) a j! f (θ) ). 1 ( 1 + ), n 1 a j+1 f ( j) (θ j 1 ) j= a j! f (θ) here c is a constant not depending on h. Thus, by (6) e get I( f ) S n ( f ) chn 1 f (n+1) (ξ) (n + 1)! f (θ) Ẽ n ( f ). 1 ( 1 + ), (9) n 1 a j+1 f ( j) (θ j 1 ) j= a j! f (θ) Therefore, for h sufficiently small, I( f ) S n ( f ) << Ẽ n ( f ), that is, Ẽ n ( f ) is a realistic correction for Q n ( f ). Furthermore, I( f ) S n ( f ) chn 1 (n + 1)! M Ẽ f (n+1) (x) n ( f ), here M = max x J f. (0) (x) Finally, by Proposition 4 and the continuity of the function f (n+1) (x), e kno that f (n+1) (ξ) (n + 1)! f [x 1, x,...,x n, x 1, x ]. So, from (9) e obtain (7). (b) The proof is analogous so it is omitted. The next proposition shos that Proposition 11 for the case n = leads to the rule S ( f ) hich is algebraically equivalent to the trapezoidal rule, and hen n = the rule S ( f ) is algebraically equivalent to the Simpson s rule. Proposition 1 Let I( f ) = b f (x)dx, J = [a, b], and f a C (J). Consider the simple extended left rectangle rule S ( f ) = Q ( f ) + Ẽ ( f ) = hf(x 1 ) + h f [x 1, x ], (1) and x = (x 1 + x )/, here x 1 = a and x = b. Assuming that f (x) 0, x (x 1, x ), 41

9 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 then Ẽ ( f ) is a realistic error for Q ( f ), for h = (b a) sufficiently small. A realistic approximation for the true error of S ( f ) is Ē S ( f ) = h f [x 1, x, x] Ẽ ( f ). () f [x 1, x ] Proof By Proposition 4 there exists a point θ (x 1, x ) such that f [x 1, x ] = f (θ), so Ẽ( f ) = h / f (θ). Using Proposition 8, e kno that I( f ) S ( f ) = I( ) f () (ξ), ξ (x 1, x ) () = h 1 f () (ξ). As by hypothesis f (θ) 0, e get I( f ) S ( f ) = h f () (ξ) Ẽ ( f ) 6 f (θ), (4) and so I( f ) S ( f ) Mh 6 Ẽ f () (x) ( f ), here M = max x [x1,x ] f. (5) (x) Therefore, for a sufficiently small h the correction Ẽ ( f ) is realistic for Q ( f ). Since f (θ) f [x 1, x] and f (ξ) f [x 1, x, x] e obtain () from (). Example 1 (A realistic error for S ( f )) Let I( f ) = xdx. The function f (x) = x is not differentiable at x = 0. Hoever f (x) 0 for x > 0, and the result () still holds. The numerical results (for 6 digits of precision) are: I( f ) = Q ( f ) = hf(0) = 0 The true error for S ( f )is By () the realistic error is Ẽ ( f ) = h f [0, 0.1] = S ( f ) = Q ( f ) + Ẽ ( f ) = E S ( f ) = I( f ) S ( f ) = Ē S ( f ) = h f [0, 0.1, 0.05] Ẽ ( f ) = f [0, 0.1] Table shos that the realistic error Ē S ( f ) becomes closer to the true error hen one goes from the step h to the step h/. Table. Realistic and true error for S ( f ) h Ē S ( f ) E S ( f ) = I( f ) S ( f ) Proposition 1 Consider the model A for n =, S ( f ) = Q ( f ) + Ẽ ( f ) = hf(x 1 ) + {h f [x 1, x ] + } h f [x (6) 1, x, x ], here x 1 = a, x = (a + b)/ and x = b. Let g(x, h) = 1 + h f () (x) 6 f (x). (7) 4

10 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 Assuming that f C 4 [a, b],if (i) f (x) 0 x (x 1, x ) (8) and (ii) g(x, h) h x (x 1, x ), (9) then, for a sufficiently small h = (b a)/, Ẽ ( f ) is a realistic correction for Q ( f ). A realistic approximation to the true error of S ( f ) is Ē S ( f ) = h f [x 1, x, x, x 1, x ] Ẽ ( f ), (40) 15 f [x 1, x ] here x 1 = (x 1 + x )/ and x = (x + x )/. Proof. As I( 4 ) = h 0 4 (t)dt = 4h 5 /15 and I( 1 ) = h 0 1 (t)dt = h, by Proposition 11 (a) e obtain (40). Example (A realistic error for the S ( f ) rule) Let I( f ) = h π e x dx = erf( h), ith h > 0. Since 0 f (x) = xe x 0, the condition (8) holds ith x 1 = 0 and x = h. Consider g(x, h) = 1 + h f () (x) 6 f (x) = 1 ( ) h x x, 0 < x < 1. As lim x 0 g(x, h) = 1 and for any 0 < x 1ehaveg(x, h) > h, for 0 < h 1 (see Figure 1), thus the condition (9) is satisfied. Notice that g(x, h) gets closer to the value 1 as h decreases. Therefore one can assure that realistic estimates (40) can be computed to approximate the true error of S ( f ) for a step h 1. In Table is displayed the estimated errors Ē S ( f ) and the true error for S ( f ), respectively for h {1/, 1/4, 1/8, 1/16}. As expected, the computed values for Ē S ( f ) have the correct sign and closely agree ith the true error g x 1 6 h 1 x 6 x h=1/ g x h=1/ x Figure 1. g(x) > h, for h = 1/, 1/4, 1/8, 1/16 For h = 1/, e have I( f ) = Q ( f ) = hf(0) = 1 Ẽ ( f ) = h f [0, 1/] + h f [0, 1/, 1] = = = S ( f ) = Q ( f ) + Ẽ ( f ) =

11 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 Table. Realistic and true error for S ( f ) h h f [x 1, x, x, x 1, x ] Ē S ( f ) = Ẽ ( f ) 15 f [x 1, x ] E S ( f ) = I( f ) S ( f ) 1/ / / / Example (A realistic error for the S 5 ( f ) rule) From Table 1 e obtain the folloing expression for the rule S 5 ( f ), { S 5 ( f ) = 4 hf(x 1 ) + 8 h f [x 1, x ] + 40 } h f [x 1, x, x ] + 16 h 4 f [x 1,...,x 4 ]. (41) Consider I( f ) = 4 h sin( x)dx = sin (4 h) and the interval J = [a, b] = [0,π/5]. Since f C 6 (J) and f (x) = 0 cos( x) 0, x J, the condition (5) holds ith x 1 = 0 and x 5 = 4 h. Let g(x, h) = 1 + a f () (x) a! f (x) + a 4 f () (x) a! f (x) + a 5 f (4) (x) a 4! f (x), a < x < b = 1 4 h (14 h 75) h tan(x), here the coefficients a i are computed using (15). It can be observed in the plot in the Figure that for h {1/8, 1/16, 1/, 1/64} the condition g(x, h) > h is satisfied. Therefore, since n is odd, one concludes from Proposition 11 (a) that the folloing realistic estimation for the true error of S 5 ( f ) is, Ē S 5 ( f ) = I( 6) f [x 1, x,...,x 5, x 1, x ] Ẽ 5 ( f ) I( 1 ) f [x 1, x ] 16 h5 f [x 1, x,...,x 5, x 1, x ] = Ẽ 5 ( f ), 1 f [x 1, x ] here x 1 = (x 1 + x )/ and x = (x 4 + x 5 )/. In Table 4 are displayed the computed realistic errors Ē S 5 ( f ) for the steps h referred above. (4) 1.0 g x h 75 h tan x 4 h 1 g x h=1/64 h=1/8 For instance for h = 1/8, e have I( f ) = I = Q 5 ( f ) = 4 hf(x 1 ) = x Figure. g(x, h) > h, for h = 1/8, 1/16, 1/, 1/64 Ẽ 5 ( f ) = 8 h f [x 1, x ] + 40 h f [x 1, x, x ] + 16h 4 f [x 1,...,x 4 ] h5 f [x 1,...,x 5 ] = = = , 44

12 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 and finally S 5 ( f ) = Q 5 ( f ) + Ẽ 5 ( f ) = Table 4. Realistic and true error for S 5 ( f ) h Ē S 5 ( f ) 16 h5 1 f [x 1, x, x, x 4, x 5, x 1, x ] f [x 1, x ] Ẽ 5 ( f ) E S 5 ( f ) = I( f ) S 5 f ) 1/ / / / Composite Rules The rules hose eights have been given in Table 1 are here applied in order to obtain the so-called composite rules. The algorithm described hereafter for composite rules is illustrated by several examples presented in paragraph 4.1. Since the best rules S n are the ones for hich n is even (hen deg(s n ) = n holds) the examples refer to S, S 5, S 7 and S 9. Whenever the conditions of Proposition 11 for obtaining realistic errors are satisfied, these rules enable the computation of high precision approximations to the integral I( f ), as ell as good approximations to the true error. This justifies the name realistic error adopted in this ork. Let n be given and fix a natural number i. Consider the number N = (n 1) i and divide the interval [a, b] into N equal parts of length h = (b a)/n, denoting by x 1, x,...,x N the nodes, ith x 1 = a, x N = b, and x j = x j 1 +h, j =,,...(N 1). Partitioning the set {x 1, x,...,x N } into subsets of n points each, and for an offset of n 1 points, e get i panels each one containing n successive nodes. To each panel e apply in succession the rule Q n and compute the respective realistic correction Ẽ n as ell as the estimated realistic error Ē S n for the rule S n ( f ). For the output e compute the sum of the partial results obtained for each panel as described in (4): Q = i k=1 Q n(panel k ) Ẽ = i k=1 Ẽn(panel k ) (4) S = Q + Ẽ (composite rule) Ē = i k=1 Ēn(panel k ) (realistic error for S ). According to Proposition 8, for composite rules ith n points by panel and step h > 0, in the favorable cases (those satisfying the hypotheses behind the theory) one can expect to be able to compute approximations S having a realistic error. Analytic proofs for realistic errors in composite rules for both models A and B ill be treated in a forthcoming ork. 4.1 Numerical Examples for Composite Rules Once computed realistic errors ithin each panel for a composite rule, e can expect the error Ē in (4) to be also realistic. This happens in all the numerical examples orked belo. Example 4 Let I( f ) = dx (Steffensen, 006, p. 161) 10 5 ln(x) = li( 10 5 ) li(10 5 ) In the interval J = [10 5, 10 5 ], the function f (x) = 1/ ln(x) belongs to the class C (J). Since f (x) = (x ln (x)) 1 0, for all x J, and for k the quotients f (k) (x)/ f (x), ith x J, are close to γ(x) = 0 and tends to the zero function γ(x) ask increases. Therefore, the lefthand side in the inequality (6) is very close to 1. That is, for n the function g(x, h) = 1 + n 1 j= a j+1 a f ( j) (x) j! f (x) is such that g(x, h) 1, for h sufficiently small. So Proposition 11 holds and one obtains realistic errors for the rules S n ( f ). 45

13 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 The behavior of the function g(x, h) is illustrated for the case n = in Figure, for h {5, 5/, 5/}. Note that in this example g(x) 1 hile the chosen steps h are greater than 1. Hoever, realistic errors are still obtained g x h 6 x h x log x 1 h=5/ g x h=5/ h= x Figure. n =. g(x) 1, for h = 5, 5/, 5/ Using a precison of decimal digits (or greater for n > ) the folloing values are obtained for the -point composite rule: I = h = 5 Q = E = = = S = Ē S = (realistic error for S ) I S = (true error). In Table 5 the realistic error is compared ith the true error, respectively for the rules ith n odd from to 9 points (the step h is as tabulated). Table 5. Comparison of the realistic error Ē S ith the true error n h Ē S I S / / / / / / / For n = 7, the computed approximation for the integral I is S = , here all the digits are correct. The simple rule S 7 ( f ) is defined (see Table 1) as S 7 ( f ) = 6 hf(x 1 ) + { 18h f [x 1, x ] + 54h f [x 1, x, x ] + 144h 4 f [x 1,...,x 4 ] h5 f [x 1,...,x 5 ] + 96h 6 f [x 1,...,x 6 ] f [x 1,...,x 7 ] }. (44) 46

14 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 The respective realistic error is (see (7)) Ē S 7 = I( 8) d 8 Ẽ 7 = 7 d i( 1 ) d 1 5 h7 8 Ẽ 7. (45) d 1 In Appendix A a Mathematica code for the composite rule S, for n = 7, is given. The respective procedure is called Q7A and the code includes comments explaining the respective algorithm. Of course e could have adopted a more efficient programming style, but our goal here is simply to illustrate the algorithm described above for the composite rules. Acknoledgments This ork has been supported by Instituto de Mecânica-IDMEC-LAETA/IST, Centro de Projecto Mecânico, through FCT (Portugal)/program POCTI. A. Appendix (Composite rule S for n = 7 points ) (* For the data {x 1,..., x n }, { f 1,..., f n }, and k 0 the output for d[x, y, k, p] is the divide difference f[x 1,..., x k ]*) (* The user may enter a precision p hich ill be assigned to the data. *) (* By default p = *) d[xi_list, yi_list, k_/; k >= 0, precision_ : In f inity] := Module[{n = Length[xi], x, y, dd}, (* set default precision to nodes xi *) x = S etprecision[xi, precision]; (* set default precision to functional values yi *) y = S etprecision[yi, precision]; (* recursive definiton for finite differences: *) dd[0, j_] := y[[ j]]; (* order 0 difference *) (* ordem i difference; dynamic computation *) dd[i_, j_] := dd[i, j] = (dd[i 1, j + 1] dd[i 1, j])/(x[[i + j]] x[[ j]]); (* Output: first difference of order k *) dd[k, 1] ]; (* The procedure q7a uses the algorithm for the simple rule ith n=7 nodes *) (* The output is a list containing the relevant items for this simple rule *) q7a[{{t1_, f 1_}, {t_, f _}, {t_, f _}, {t4_, f 4_}, {t5_, f 5_}, {t6_, f 6_}, {t7_, f 7_}}, precision_ : In f inity]:= Module[{x1, x, x, x4, x5, x6, x7, xb1, xb, y1, y, y, y4, y5, y6, y7, yext, hh, ext, d1, d8, q, e1, e, e, e4, e5, e6, E7, s, real}, {y1, y, y, y4, y5, y6, y7} = S etprecision[{ f 1, f, f, f 4, f 5, f 6, f 7}, precision]; {x1, x, x, x4, x5, x6, x7} = S etprecision[{t1, t, t, t4, t5, t6, t7}, precision]; xb1 = (x1 + x)/; xb = (x6 + x7)/; ext = {x1, x, x, x4, x5, x6, x7, xb1, xb}; yext = Map[ f, ext]; (* completation of the panel *) d1 = d[{x1, x}, {y1, y}, 1]; (* divided difference order 1 *) d8 = d[ext, yext, 8, precision]; (* divided difference order 8 *) hh = S etprecision[h, precision]; (* step *) q = 6 hh y1; (* left rectangle quadrature rule *) e1 = 18 hh d[{x1, x}, {y1, y}, 1]; (* error e1: *) e = 54 hh d[{x1, x, x}, {y1, y, y}, ]; (* error e: *) 47

15 .ccsenet.org/jmr Journal of Mathematics Research Vol. 4, No. 5; 01 e = 144 hh 4 d[{x1, x, x, x4}, {y1, y, y, y4}, ]; (* error e: *) e4 = 1476/5 hh 5 d[{x1, x, x, x4, x5}, {y1, y, y, y4, y5}, 4]; (* error e4: *) e5 = 96 hh 6 d[{x1, x, x, x4, x5, x6}, {y1, y, y, y4, y5, y6}, 5]; (* error e5: *) e6 = 1476 hh 7 /7 d[{x1, x, x, x4, x5, x6, x7}, {y1, y, y, y4, y5, y6, y7}, 6]; (* error e6: *) E7 = e1 + e + e + e4 + e5 + e6; (* realistic error for q *) s = q + E7; (* s is a realistic approximation to the integral *) real = N[ 7 hh 7 /5 d8/d1 E7, 8]; (* realistic error for s *) {q, e1, e, e, e4, e5, e6, E7, s, real}]; (* output *) (* The folloing procedure Q7A gives a list containing the relevant items for the composite rule. It calls the procedure q7a: *) Q7A[x_List, y_list, precision_ : In f inity] := Module[{data, list}, (* partition into 7 nodes offset 6 *) data = Partition[Transpose[{x, y}], 7, 6]; (* sum entries in cells and prepend the step h used *) list = Map[q7A[#, precision]&, data]; Prepend[Map[Apply[Plus, #]&, Transpose[list]], Rationalize[h]]]; References Davis, P. J., & Rabinoitz, P. (1984). Methods of Numerical Integration. Orlando: Academic Press. Gautschi, W. (1997). Numerical Analysis, An Introduction. Boston: Birkhäuser. Graça, M. M. (01). Realistic errors for corrected open Neton-Cotes rules. (In preparation). Isaacson, E., & Keller, H. B. (1966). Analysis of numerical methods. Ne York: John Wiley & Sons. Krylov, V. I. (005). Approximate Calculation of Integrals. Ne York: Dover. Steffensen, J. F. (006). Interpolation (nd ed.). Boston: Dover. 48